Extending Dynamics to Tree Graph Domains: Two Examples from...
Transcript of Extending Dynamics to Tree Graph Domains: Two Examples from...
Extending Dynamics to Tree Graph Domains: Two Examples
from Math Biology
Jon Bell, UMBC
1. Cable theory and dendritic trees
(Pyramidal cell from mouse cortex; by Santiago Ramon)
2. Population persistence in advection-dominated environments (river
networks)
(Amazon river basin: courtesy of Hideki Takayusu)
(Purkinje cells, fluorenscent dyed;
Technology Review, Dec. 2009)
−
+∂
∂=+
∂
∂=
∂
∂∑
−
m
jion
jj
mionm
i
R
v
Evwg
t
vCvI
t
vC
x
v
R
a))((
),(2 2
2
K
Example dynamics for this talk: bistable equation
)(2
2
ufx
u
t
u+
∂
∂=
∂
∂
>
<<>
<<<
==
>∈
=
∫1
0
1
0)(
10)(
00)(
0)1()0(
1],0[
:
dssf
vforvf
vforvf
ff
AACf
H f
α
α
Neuronal Cable Theory on a Dendritic Tree
(metric tree graph)
outside
inside
membrane
0hr0
hr
ihr ihr
)(0 hxv − )(0 hxv +
)( hxvi +)( hxvi − )(xvi
)(0 xv
)(xhim)( hxhim
−
)( hxhiext − )( hxhiext +)(xhiext
Problem: (1) )(2
2
ufx
u
t
u+
∂
∂=
∂
∂ on ),0(\ TV ×Ω
(2) 0=u on 0×Ω
(3) 0),(~
=∂∑ν
ν
je
j tu for 1\ γν V∈ (Kirchhoff-Neumann)
u is continuous at ν
(4) 0)(),( 11 >=∂− tItu γ for ],0[ Tt ∈
1fH : There exists 0, 00 >fu such that for 00 uu ≤≤ , ufuf 0)( −≤ .
Let ( ) ( )],0(\],0[: 1,2TVCTCZ
T×Ω∩×Ω= .
Theorem: Let f satisfy 1, ff HH . Let T
Zu ∈ be a solution to (1)-(4), for any T
> 0, where
≥
<<=
−−
0
)(
1
00
0
0)(
ttforeI
ttforItI
ttδ . Here 010 0,0 fII <<>≥ δ . Then
0),(lim =∞→
txu jt
for all jex ∈ , all j = 1,…,N.
2fH : for some 0>bf , ufuf b−≥)( for 0≥u .
Theorem: Let f satisfy 1, ff HH . Let TZu ∈ solve (1)-(4) with )(tI replaced by
)(* tIµ , where 0)(* ≥tI , *I not identically zero. Then there exists a 0µ
depending on *I , such that if 0µµ > , for each j, ),(),( txvtxu jj ≥ and
1),(lim =∞→
txv jt
.
VE ∪=Ω
MN VeeeE ννν ,...,,,...,, 2,121 ==
mindexV γγγνν ,...,,1)(| 21==∈=Ω∂
2)(|\ >∈=Ω∂ νν indexVV
=Ω metric graph if every edge Ee j ∈ is
identified with an interval of the real
line with positive length jl .
=Ω tree graph if there are no cycles.
Bounds on the Speed of Propagation: IVP example
Consider the problem in ),0( ∞×Ω∞ :
Assume u is a solution to (IVP) such that
(*) 1lim =∞→
jt
u in ),0( ∞×Ω∞
Write )()()0(')( ugauugufuf +−=+= , g is smooth, g(u) = O(u2) as
0→u . Define 0/)(sup:10
>=<<
uugu
σ .
Note that L 0)(: ≤−=−+−= jjjjjxxjtj uuguauuuu σσ .
Assume 3,2,1,10 =≤≤ jjφ . Then (by another comparison result),
10 ≤≤ jv .
Theorem: Suppose v is a solution to (IVP) satisfying (*). Let φ have
bounded support, supp 1e⊂φ . If σσ +=> acc /2: , then for each j, each
jex ∈ , 0),(lim =+∞→
tctxu jt
.
Suppose (IVP) admits a positive steady state solution q(x) for
0<≤≤ bxa , with q(a) = q(b) = 0. Suppose the only nonnegative global
steady state τ of (IVP) with )()(1 xqx ≥τ on [a,b] is 1≡τ . Let u be a
solution to (IVP) with )()(1 xqx ≥φ on [a,b] .
Then there is a 0>c such that for cc <<0 , for any x, any 0>ε , there
is a T > 0 such that for t > T, 3,2,1,1),( =−≥+ jtctxu j ε .
(IVP) =
>+
×Ω=
×Ω+=
0:
0
),0(\)(
tatcontinuityKN
inu
TVinufuu xxt
γ
φ
Inverse Problem
Motivation:
Problem: (1) xxt uuxqu =+ )( on ),0(\ TV ×Ω
(2) 0=u on 0×Ω
(3) 0),(~
=∂∑ν
ν
je
j tu for Ω∂∈ \Vν
u is continuous at Ω∂∈ \Vν
(4) )],,0([2 m
TLfu ℜ∈=∂ on ],0[ T×Ω∂
Strategy: determine q(x) on boundary edges via Boundary Control Theory,
then “prune” tree to smaller tree.
Reference: Avdonin & Bell, JMB, 2012; Avdonin & Bell, submitted, Inverse
Problems
Another Problem: xxt uuxquxq =++ )())(1(
From Mcgee and Johnston, J. Physiol., 1988; and
Traub, et al, J. Neurophys., 1991
Some Questions/Remarks
Subprojects:
1. Tree with variable edge diameters: conduction block
Let ϕ=u on 0×Ω∞ , so );,( ϕtxu is a solution on ],0( T×Ω
∞. For a class
A of admissible i.c.s, let ℜ→Ω∞
:, 21 ψψ have property that
21 );,( ψϕψ ≤⋅≤ tu in ],0[ T×Ω∞ . Then 21,ψψ will be a trap for A. );,( ϕtxu
will be blocked at γ if there is a trap 21,ψψ such that )(|||| ερψ ≤i ,
2,1=i and 0)( →ερ as +→ 0ε .
2. Distal current sources: Given stimulus )(tI , voltage boundary conditions on
1\ γΩ∂ , and sealed-end conditions, can we estimate 1u at 1γ (representing
potential at the soma)?
3. We have (weak) threshold conditions for FHN system on ),0( ∞×Ω :
wuwwufuu txxt µσ −=−+= ,)(
4. Want to show existence of a traveling wave solution on ],0( T×Ω∞ .
5. Want to explore (numerically) other properties, e.g., wave attenuation and
amplification along graph domains.
Query: How do populations persist in face of downstream biased flow? (The
“drift paradox”; Muller, 1954; Hershey, et al, 1993, …)
Suggestions: drifting populations represent a surplus that would not
otherwise contribute to local persistence, or compensatory upstream
movements (colonization cycle).
Lutscher, et al, 2010: populations must have ability to invade upstream in
order to persist.
Usual modeling: involves an advection-diffusion-reaction equation on a
finite interval (single river branch), or on the real line.
References above initiated persistence investigations on a branching
(metric graph) domain.
Population Persistence
in River Networks
(metric tree graph)
References:
Sarhad, Carlson, Anderson, JMB,
2012
Ramerez, JMB, 2012
Vasilyeva, Lutscher, BMB, 2012
(1) )(ufsuDuu xxxt +−= in ),0(\ ∞×Ω V
u = population density s = downstream advection speed (population
drift), which may differ from rate of current
due to organism behavior
)(uf = birth/death process. Here let 0,)( >= rruuf
Flux on
+
∂
∂−=⋅= jj
j
jjjj usx
uDAtqe ),( , jA = cross-sectional area of segment
Simplifying model assumptions: jallssDD jj _,, ==
x increases downstream
Continuity at vertex ν : 0,210 ≥== tuuu
Conservation at vertex ν : ),(),(),( 210 tqtqtq ννν +=
At interior vertex ν , incident sectional area difference is 210 AAAB −−=ν .
Assume
(2) 0)( 210 ≥−−= sAAAsBν for Ω∂∈ \Vν
Upstream boundary vertex condition: Ω∂∈= γγ ,0),( tq
Inhospitable river ending at downstream vertex dγ : 0),( =tu dγ
Reduction: ⇒= ),(),( 2/ txwetxu Dsx
(3) 0)4
(2
=−+= wD
srDww xxt in ),0(\ ∞×Ω V
(4) at dγγ \Ω∂∈ : 02
=
−
∂
∂= w
s
x
wDAq j ( )γ~je
(5) at dγ : 0),( =tw dγ
(6) at Ω∂∈ \Vν :
∂
∂−+
∂
∂−=
∂
∂−
x
wDw
sA
x
wDw
sA
x
wDw
sA 2
221
110
00222
Remark: On V\Ω , set 04
2
=−D
sr , ⇒∂
∂=
2
2
:x
DL Lwwt = is formally
self-adjoint. With Ω a metric tree graph with finitely many edges, L
can be shown self-adjoint with compact resolvent, so there is a
spectrum jλ , −∞→jλ ; and an ON basis jϕ of eigenfunctions. When
(2) 0)( 210 ≥−−= sAAAsBν at each Ω∂∈ \Vν
w equation satisfies a maximum principle. If ),( txu on Ω is continuous,
non-negative, then 0),( ≥txu for all 0>t .
Theorem: Suppose w satisfies (3)-(6), with (2) holding. Then, if
||4 1
2
λ<−D
sr , the population will not persist. (In particular, it will not
persist if 04
2
≤−D
sr .) If ||4 1
2
λ≥−D
sr a continuous positive initial
population will persist.
Remarks:
1. For the linear advection-diffusion-reaction equation on a finite
interval , with no flux at 0=x , homogeneous Dirichlet condition (hostile
river ending) at Lx = , it is an undergraduate exercise to show no
persistence (i.e., population decreases) if 1
24/ λ−≤− Dsr (and growth if
1
24/ λ−>− Dsr ).
2. On a single branch, with
)1( urusuDuu xxxt −+−= on 0,0 ><< tLx
( ) 0,0),(,0),0( >==+− ttLutsuDu xx
Vasilyeva & Lutscher, 2012 show there is no positive steady state if
04/2≤− Dsr (no persistence). For each sD, and 04/2
>− Dsr there
exists a ),(** sDLL = such that there is a unique positive steady state if
*LL > , and it is stable. We are pushing the Vasilyeva-Lutscher work to
the metric tree graph setting, using
))(1()( α−−= uuruuf
Which encompasses an allee-type mechanism.
3. Further out in consideration is to explore competition in a network
advection-driven environment; maybe with basic equations
−−+−=
−−+−=
)(
)1(
2
1
buvrvsvvDv
avuusuuDu
xxxt
xxxt
4. Return to a single population, but consider parameters that vary
depending on branch, and investigate different downstream river
ending boundary conditions.
From A. Rinaldo, et al, PNAS, 109(17), 2012.
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